7.5-Determinants in Two Variables

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1 7.-eteminnts in Two Vibles efinition of eteminnt The deteminnt of sque mti is el numbe ssocited with the mti. Eve sque mti hs deteminnt. The deteminnt of mti is the single ent of the mti. The deteminnt of mti is defined s follows: The deteminnt of the mti,,,, is the el numbe In smbols,,,,, Given mti A, the deteminnt of the mti is witten det(a). An es w to emembe how to find the deteminnt of mti is tht the numbe is found b subtcting the poduct of the non-min digonl enties fom the poduct of the min digonl enties. Emple: Find the deteminnt of mti A. A 4 Solution: Subtct the poduct of the non-min digonl enties fom the poduct of the min digonl enties. 4 4 Theefoe, det( A ) Emple: Find the deteminnt of mti B. B 4 Solution: Subtct the poduct of the non-min digonl enties fom the poduct of the min digonl enties. 4 ( )( ) ()(4) Theefoe, det( B )

2 Cme s Rule fo Sstems in Two Vibles: We will now see how deteminnts ise in the solution of sstem of line equtions in two unknowns. Given the sstem of two line equtions in two vibles: Then b c b c nd det is the deteminnt obtined fom the coefficients of the sstem. b b c b det is obtined b eplcing the -coefficients b the constnts. c b det is obtined b eplcing the -coefficients b the constnts. c c Emple: Use Cme s Rule to solve the sstem: 4 Solution: Find, nd. det b b det 4 ()( ) ()( 4) 4 c det c b b det 4 ()( ) ()( 4) 4 det c c det ()() ()() 7 Then: 7 7 The solution is (, 7).

3 eteminnts: Inconsistent nd ependent Sstems:. If nd t lest one of the deteminnts in the numeto is not, then the sstem is inconsistent nd thee is no solution.. If nd ll the deteminnts in the numeto e, then the equtions e dependent nd thee e infinitel mn solutions. Emple: Use Cme s Rule to solve the sstem: Solution: Find, nd. det ()() ( )( ) det ()() ()( ) det ()() ( )() Since nd t lest one of the deteminnts in the numeto e not, the sstem is inconsistent. Emple: Use Cme s Rule to solve the sstem: Solution: Find, nd. det ()() (4)() det (4)() (8)() det ()(8) (4)(4) 4 8 Since nd ll the deteminnts in the numeto e, the sstem is dependent.

4 Eigenvlues nd Eigenvectos Suppose tht A is sque mti of ode n n nd V is vecto mti. If thee is numbe such tht AV V, then we s tht is n eigenvlue nd V is n eigenvecto of mti A. Emple: Given mti A nd vecto mti V below, show tht is n eigenvlue nd V is n eigenvecto of A. A 4 V Solution: In ode fo to be n eigenvlue nd V to be n eigenvecto, the must solve the mti eqution AV V. AV V 4 Theefoe, is n eigenvlue nd V is n eigenvecto of mti A.

5 etemining Eigenvlues Finding eigenvlues is mtte of solving polnomil eqution. Let A be sque mti of ode n n. If the numbe is n eigenvlue of A, then det( A I ) If we think of s vible, then det( A I ) is polnomil of degee n clled the chcteistic polnomil. The chcteistic eqution is the eqution det( A I ). 7 8 Emple: Find ll the eigenvlues of the mti A. 8 Solution: We need to solve the chcteistic eqution det( A I ). To do this, fist find A I A I A I A I Now we cn solve det( A I ) det( A I) 7 det 8 8 ( 7 )(8 ) ( )(8) This is the chcteistic eqution. Solve the eqution fo. ( )( ) Theefoe, the eigenvlues e -, nd.

6 etemining Eigenvectos Once we know the eigenvlue, we cn find the coesponding eigenvectos b solving the mti eqution V AV fo V. Fist simplif the eqution s follows. ) ( V I A V AV V AV whee the eigenvectos V ssocited with the eigenvlues e the non-zeo solutions to this eqution. Emple: Find the eigenvectos coesponding to ech eigenvlue of mti A, given tht the eigenvlues e -, nd A Solution: Since ech eigenvlue hs its own eigenvecto, we must find two eigenvectos. Fo ech eigenvecto, we need to solve the mti eqution ( ) V I A Fist: Solve the mti eqution ( ) V I A when

7 To solve this sstem, we must ewite it s n ugmented mti nd use Guss-Jodn Elimintion. 8 R R R R R NOTE: The bottom ow must be ll zeos indicting tht the sstem is dependent nd thee e n infinite numbe of solutions. Consequentl, eve eigenvlue hs n infinite numbe of eigenvectos. An eigenvecto of the fom will be n eigenvecto fo the eigenvlue of Theefoe v is n eigenvecto fo the eigenvlue. Second: Solve the mti eqution ( ) V I A when ) (

8 To solve this sstem, we must ewite it s n ugmented mti nd use Guss-Jodn Elimintion. 8 R R R R R An eigenvecto of the fom will be n eigenvecto fo the eigenvlue of - Theefoe v is n eigenvecto fo the eigenvlue -. Checking the Solutions: We cn now show tht the pis of eigenvlues nd eigenvectos do indeed solve the mti eqution AV V when 7 8 A 8 Fo the eigenlue/eigenvecto pis: nd v v AV V AV V

9 .- Applictions Gowth Rtes & Eigenvlues One ppliction of eigenvlues is fo pedicting popultion gowth. We efe to the lgest eigenvlue of mti s the long tem gowth te. To find the long tem pecentge gowth te we subtct fom the long tem gowth te nd wite this diffeence s pecentge. Emple: The Leslie mti fo ovenbids in centl Missoui is given b: L Whee goup epesents htchlings nd goup epesents dults. Find the long tem gowth te nd the long tem pecentge gowth te. Solution: The long tem gowth te is the lgest eigenvlue of the Leslie mti. To find the eigenvlues we need to solve the chcteistic eqution det( A I ). To do this, fist find A I A I A I A I Now we cn find the det( A I ) det( A I ).78. det.. (.78 )(. ) (.)(.).48.7 This is the chcteistic eqution. Solve the eqution fo..48 ±..48 (.48) () 4()(.7) The long tem gowth te is the lgest eigenvlue which is.48. The long tem pecentge gowth te is %.

10 Gowth Rtes & Eigenvectos Eigenvlues nd eigenvectos hve mn pplictions in science. The following emples will demonstte how the m be used to pedict futue gowth tes of popultion. Suppose tht the Leslie mti L fo popultion hs eigenvectos eigenvlues m v, v v Λ v ssocited with,, Λ espectivel. If the initil popultion vecto is, v v v Λ v m m Then fte n time peiods, the popultion pojection vecto is m n n n n v v v Λ v n m m m Emple: The Leslie mti fo popultion of htchlings nd dult bids hs the following eigenvlues nd eigenvectos. The initil popultion vecto is lso given. v v 4 Wite the initil popultion vecto s line combintion of eigenvectos. Find the popultion fte es. Solution: Find the vlues of nd b using the initil popultion vecto nd given vlues to cete mti eqution. v v 4 4 We must now detemine the vlue of nd. This mti eqution cn be solved b n method. 4 4 Substitute these vlues into the bove mti eqution to obtin the initil popultion vecto s line combintion of eigenvectos. 4 Use the popultion pojection vecto nd given vlues to cete mti eqution. The popultion vecto fte es is: v v

11 () Theefoe, the numbe of htchlings nd dults in e will be 4,8 nd,4 espectivel. ocedue fo Solving opultion ojection oblems: In the net emples it will be necess to detemine ll the equied vlues; eigenvlues, eigenvectos, nd vlues. The following pocedue will be ecommended.. Find ll eigenvlues.. Find eigenvecto fo ech eigenvlue.. Use initil popultion vecto to detemine the " vlues. 4. Use the popultion pojection vecto to detemine the futue popultion. Emple: Clculte the popultion of wild geese in e given the following initil popultion vecto nd Leslie Mti L. Age goups e htchlings nd dult. L Solution: st Step: Find the eigenvlues. To do this we need to solve the chcteistic eqution det( L I ). Fist find L I L I L I L I Now we cn find the det( L I )

12 4 4 ()() ) )( ( det This is the chcteistic eqution. Solve the eqution fo. 4 ) 4( ( 4 4 Theefoe, the eigenvlues e 4, nd. nd Step: Find the eigenvecto fo ech eigenvlue. Fo ech eigenvecto, we need to solve the mti eqution ( ) v I L Find the eigenvecto fo the eigenvlue Solve this sstem of equtions b Gussin elimintion.. R R R An eigenvecto of the fom will be n eigenvecto fo the eigenvlue of 4. Theefoe v is n eigenvecto fo the eigenvlue 4.

13 Find the eigenvecto fo the eigenvlue. To solve this mti eqution using Gussin elimintion. R R R An eigenvecto of the fom will be n eigenvecto fo the eigenvlue of. Theefoe v is n eigenvecto fo the eigenvlue. d Step: Use the initil popultion eqution to detemine the vlues. 4 v v Substitute the eigenvectos into the initil popultion eqution long with the initil popultion vecto nd solve fo the vlues. v v 4

14 Theefoe, the vlues e: 4 4th Step: Use the popultion pojection vecto to detemine the futue popultion. Substitute ll vlues into the eqution nd solve fo vecto. v v 4(4) () 7,88,4 7,88,4 7,88,4,,,,,,,,,,,,,,8,,,,7,88,4 In e thee will be,,8,, htchlings nd,,7,88,4 dults.

Topics for Review for Final Exam in Calculus 16A

Topics for Review for Final Exam in Calculus 16A Topics fo Review fo Finl Em in Clculus 16A Instucto: Zvezdelin Stnkov Contents 1. Definitions 1. Theoems nd Poblem Solving Techniques 1 3. Eecises to Review 5 4. Chet Sheet 5 1. Definitions Undestnd the